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Astronomy & Astrophysics manuscript no. paper c©ESO 2018May 8, 2018

Dynamo cycles in global convection simulations of solar-like stars

J. Warnecke1

Max-Planck-Institut für Sonnensystemforschung, Justus-von-Liebig-Weg 3, D-37077 Göttingen, Germanye-mail: warnecke@mps.mpg.de

May 8, 2018

ABSTRACT

Context. Several solar-like stars exhibit cyclic magnetic activity similar to the Sun as found in photospheric and chromosphericemission.Aims. We want to understand the physical mechanism involved in rotational dependence of these activity cycle periods.Methods. We use three-dimensional magnetohydrodynamical simulations of global convective dynamos models of solar-like stars toinvestigate the rotational dependency of dynamos. We further apply the test-field method to determine the α effect in these simulations.Results. We find dynamo with clear oscillating mean magnetic fields for moderately and rapidly rotating runs. For slower rotation,the field is constant or exhibit irregular cycles. In the moderately and rapidly rotating regime the cycle periods increase weakly withrotation. This behavior can be well explained with a Parker-Yoshimura dynamo wave traveling equatorward. Even though the α effectbecomes stronger for increasing rotation, the shear decreases steeper, causing this weak dependence on rotation. Similar as othernumerical studies, we find no indication of activity branches as suggested by Brandenburg et al. (1998). However, our simulationseems to agree more with the transitional branch suggested by Distefano et al. (2017) and Olspert et al. (2017). If the Sun exhibit adynamo wave similar as we find in our simulations, it would operate deep inside the convection zone.

Key words. Magnetohydrodynamics (MHD) – turbulence – dynamo – Sun: magnetic fields – stars: activity – stars: magnetic fields

1. Introduction

The Sun, our nearest late-type star exhibits a magnetic activ-ity cycle with a period of around 11 yrs. The cyclic mag-netic field is generated by a dynamo operating below thesurface, where it converts the energy of rotating convectiveturbulence into magnetic energy. The solar dynamo mecha-nism is still far from being fully understood (e.g. Ossendrijver2003; Charbonneau 2014). One reason is the limited infor-mation about the dynamics in the solar convection zone. He-lioseismology have provided us with the profile of tempera-ture and density stratification and the differential rotation (e.g.Schou et al. 1998) in the interior, further information such asthe meridional circulation profiles, convective velocity strengthor even magnetic field distributions are currently inconclusiveor not even possible (e.g. Basu 2016; Hanasoge et al. 2016).One way to investigate how important differential rotation,meridional circulation and turbulent convective velocities arefor the solar dynamo, is to use numerical simulations. Sincethe early simulations by Gilman (1983), there have been sev-eral advances using numerical simulation due to the increaseof computing resources. Nowadays, global simulations of con-vective dynamos are able to reproduce cyclic magnetic fieldsand dynamo solutions resembling many features of the so-lar magnetic field evolution (Ghizaru et al. 2010; Käpylä et al.2012; Warnecke et al. 2014; Augustson et al. 2015), even thelong-time evolution (Augustson et al. 2015; Käpylä et al. 2016;Beaudoin et al. 2016). The cyclic magnetic field in these simula-tions can be well understood in terms of Parker-Yoshimura rule(Parker 1955; Yoshimura 1975; Warnecke et al. 2014), where apropagating αΩ dynamo wave is excited, see also Gastine et al.(2012). The α effect (Steenbeck et al. 1966) describes the mag-netic field enhancement from helical turbulence and the Ω effect

the shearing of magnetic field caused by the differential rota-tion. The propagation direction of the dynamo wave depends onthe sign of α and shear: for generating an equatorward propa-gating wave, the product of α and the radial gradient of Ω mustbe negative (positive) in the northern (southern) hemisphere. Forexplaining the solar equatorward propagation of the sunspot ap-pearance by the Parker-Yoshimura rule therefore requires eitherto invoke the near-surface-shear layer (Brandenburg 2005), be-cause only there the radial gradient is negative (Barekat et al.2014) and α is positive or that α changes sign in the bulk ofthe convection zone (Duarte et al. 2016), where the radial shearis positive. Furthermore, to fully understand the magnetic fieldevolution in the global numerical simulation one needs suit-able analysis tools to extract the important contribution of turbu-lent dynamo effects. One of these tools is the test-field method(Schrinner et al. 2005, 2007; Warnecke et al. 2018). This methodallows to determine the turbulent transport coefficients directlyfrom the simulations. This include the measurement of tensorialcoefficients such as α, turbulent pumping and turbulent diffusion.Already the first application to global convection simulation ofsolar-like dynamo revealed that the turbulent effects can havea significant impact on the large-scale magnetic field dynamics(Warnecke et al. 2018; Gent et al. 2017).

Another possibility to understand the solar dynamo makeuse of the observation of other stars. Since the Mount Wilsonsurvey, we know that many stars exhibit cyclic magnetic ac-tivity (e.g. Noyes et al. 1984a,b; Baliunas et al. 1995). In thissurvey, they observe solar-like stars in the chromospheric Ca IIH&K band, which is used as a proxy for magnetic activity. Us-ing this data Brandenburg et al. (1998) and Saar & Brandenburg(1999) found two distinguish branches, when they plot theratio of rotational period and activity cycle period Prot/Pcycl

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over the rotational influence on the stellar convection in termsof the inverse Rossby number. The two branches are calledinactive and active branch, because of their preferred mag-netic activity, divided by the so-called Vaughan–Preston gap(Vaughan & Preston 1980). Their slopes are positive in termsof rotational influence, this means the cycle period decreasesfaster than linear with increasing rotation. This agrees qualita-tively with the finding of Noyes et al. (1984b), where they obtainPcycl ∝ P1.25

rot . However, Oláh et al. (2016) find in their recent re-

analysis of the Mount Wilson data a relation of Pcycl ∝ P0.24rot .

The activity branches of Brandenburg et al. (1998) have been re-cently supported (Brandenburg et al. 2017), but also questioned(Reinhold et al. 2017; Distefano et al. 2017; Boro Saikia et al.2018; Olspert et al. 2017). One of the short comings is clearlythe use of the ill-determine convective turnover time τc, whichis used to the calculated the Rossby number Ro = 4πProt/τc; forevery star τc is highly depth-dependent and different location ofa dynamo might invoke a different τc. However, these branchescan be also obtained using the fractional Ca II H&K emissionR′

HKinstead of the Rossby number (see e.g. Brandenburg et al.

2017; Olspert et al. 2017).

Explaining the observational findings via dynamo mod-els has been challenging. Simple mean-field models of tur-bulent dynamos produce rotational dependencies of cycle pe-riod similar to the observed ones using overlapping induc-tion layers (Kleeorin et al. 1983). Advective dominated fluxtransport models tend to produce an increase of cycle periodswith increasing rotation rate (e.g. Dikpati & Charbonneau 1999;Bonanno et al. 2002; Jouve et al. 2010), which is opposite whatis observed. In these models the cycle length is mainly de-termined by the strength of return flow of the meridional cir-culation, which is believed to decreases with increasing rota-tion (e.g. Köhler 1970; Brown et al. 2008; Warnecke et al. 2016;Käpylä et al. 2017; Viviani et al. 2018). However, the models ofKitchatinov & Rüdiger (1999) show an increase of meridionalcirculation strength with rotation, leading to a decrease of cycleperiod with rotation (e.g. Küker et al. 2001), which agrees qual-itatively with observations.

Another possibility to explain cycles in dynamo models isvia the turbulent (eddy) magnetic diffusivity. In a propagat-ing αΩ dynamo wave, the dynamo drivers, whose are respon-sible for the cycle length, have to balance with the contribu-tion from the turbulent diffusion. Using a turbulent diffusivityof ηt = 2 × 108 m2/s one gets a magnetic cycle length of around23 yrs (e.g. Roberts & Stix 1972), which is pretty close to solarvalue of 22 yrs. A change in the cycle length can be then associ-ated with a change in the turbulent diffusion caused by magneticor rotational quenching (e.g. Rüdiger et al. 1994). These authorsfound a cycle dependence of Pcycl ∝ P0.1

rot .

There has been only a limited number of studies of rotationaldependencies of dynamo cycles using global dynamo simula-tion. Strugarek et al. (2017) found Pcycl ∝ P−1.06

rot in a rather lim-ited sample of rotation rates. In the recent work by Viviani et al.(2018), the author found no clear dependency of cycle periodswith rotation. However, a decrease in cycle period with increas-ing rotation seems be more likely than an increase, which mightbe because of the strong oscillatory non-axisymmetric magneticfields in these simulations.

In this work, we present the results of spherical convec-tive dynamo models with rotation rates varying by a factor of30. We will determine the cycle dependency on rotational in-fluence, see Section 3.1, interpreted the data in terms of theParker-Yoshimura rule by using test-field obtained transport co-

efficients, see Section 3.2, and compare the finding with obser-vational as well as other numerical results in Section 3.3.

2. Model and setup

The detailed description of the general model can be found inKäpylä et al. (2013) and will not be repeated here. We modelthe convection zone of a solar-like star in spherical polar co-ordinates (r, θ, φ) using the wedge assumption 0.7 R < r < R,Θ0 < θ < π−Θ0 and 0 < φ < π/2 with R being the stellar radiusand Θ0 = 15. We solve the evolution equations of compress-ible magnetohydrodynamics for the magnetic vector potential A,which therefore ensures the solenoidality of the magnetic fieldB = ∇ × A, for the velocity u, the specific entropy s and densityρ. The model assumes an ideal gas for the equation of state. Thefluid is also influenced by Keplerian gravity and rotation via theCoriolis force. Because of the wedge assumption, we use peri-odic boundary condition in the azimuthal (φ) direction. We as-sume a stress-free condition for the velocity field at other bound-aries and perfect conducting latitudinal and bottom boundariesand radial field condition at the top boundary for the magneticfield. The energy is transported into the system via a constantheat flux at bottom boundary and the temperature obeys a blackbody condition. On the latitudinal boundaries, the energy flux isvanishing using zero derivative for the thermal-dynamical quan-tities. The detailed setup including the exact equations and ex-pression for the boundary condition can be found in Käpylä et al.(2013, 2017) and Warnecke et al. (2014).

We characterize our runs with the following non-dimensional input parameters; the Taylor number, the SGS, andmagnetic Prandtl numbers

Ta = [2Ω0(0.3R)2/ν]2, PrSGS =ν

χSGSm

, PrM =ν

η, (1)

where ν and η are the constant kinematic viscosity and mag-netic diffusivity, and the sub-grid-scale (SGS) heat diffusivityχSGS

m = χSGS(rm) is evaluated at rm = 0.85R. Furthermore, weuse the Rayleigh number obtained from the hydrostatic stratifi-cation, evolving a 1D model, given by

Ra=GM(0.3R)4

νχSGSm R2

(

−1

cP

dshs

dr

)

(r=0.85R)

, (2)

where shs is the hydrostatic entropy. As diagnostic parameters,we quote the density contrast.

Γρ ≡ ρ(r = 0.7R)/ρ(R), (3)

the fluid and magnetic Reynolds numbers and the Péclet number,

Re =urms

νkf

, ReM =urms

ηkf

, Pe =urms

χSGSm kf

, (4)

where kf = 2π/0.3R ≈ 21/R is an estimate of the wavenumberof the largest eddies. We defined the Coriolis number as

Co = 2Ω0τc ≡ Ro−1, (5)

where τc = 1/urmskf is the convective turnover time and urms =√

(3/2)〈u2r + u2

θ〉rθφt is the rms velocity and the subscripts indi-

cate averaging over r, θ, φ and a time interval covering the sat-urated state. The duration of the saturated state is indicated byτsat and covers several magnetic diffusion times. The kinetic andmagnetic energy density are given by

Ekin =1

2〈ρu2〉rθφt, Emag =

1

2µ0

〈B2〉rθφt. (6)

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Warnecke: Dynamo cycles in global convection simulation

Table 1. Summary of Runs.

Run Ω Ta[106] Ra[107] Re Co Pcycl[yr] EPcycl[yr] Pcycl[yr] EPcycl

[yr] PPY[yr] Emag/Ekin τsat[yr]

M0.5 0.5 1.3 4.0 44 0.7 1.6 1.3 3.9 12.4 2.7 0.06 68M1 1.0 5.4 4.0 40 1.5 20.4 9.3 34.7 19.9 2.1 0.16 104M1.5 1.5 12 4.0 39 2.2 23.1 9.4 46.2 32.6 2.3 0.17 92M2 2.0 22 4.0 40 2.9 10.8 2.4 33.1 23.4 2.2 0.10 66M2.5 2.5 34 4.0 40 3.7 12.1 4.1 7.7 16.3 1.8 0.10 61M3 3.0 49 4.0 39 4.5 6.4 0.7 4.6 19.5 2.1 0.13 64M4 4.0 86 4.0 36 6.6 2.4 0.1 2.6 0.3 1.6 0.21 47M5 5.0 35 4.0 34 8.6 2.2 0.1 2.3 0.3 1.9 0.29 70M7 7.0 264 4.0 31 13.4 2.6 0.1 2.7 0.5 2.4 0.39 51M10 10.0 540 4.0 27 21.5 2.7 0.2 2.5 16.9 2.7 0.52 53M15 15.0 1897 7.4 27 40.3 3.8 0.2 3.8 18.8 4.2 0.85 61

Notes. Second to fourths columns: input parameters. Last ninth columns: diagnostics computed from the saturated states of the simulations. τsat

indicate the time span of the saturated stage. Pcycl and Pcycl are the cycle periods determined from the magnetic field components and the magnetic

energy, respectively, EPcycland EPcycl

are there corresponding error estimates, see Section 3.1. PPY are the cycle periods determined using a Parker-Yoshimura dynamo wave, see Equations (7) and (8) and Section 3.2. Emag/Ekin is ratio of magnetic to kinetic energy. All runs have PrSGS = 2 andPrM = 1 and a density contrast of Γρ = 31.

All values for these non-dimensional input and diagnostic pa-rameters are shown in Table 1 for all runs.

The wedge assumption in the azimuthal (φ) direction allowsus to suppress non-axisymmetric dynamo mode with azimuthaldegree m = 1, 2, 3 and therefore use the mean-field decomposi-tion to describe the large-scale velocity and magnetic field. Weuse an over-bar to refer to the mean, azimuthal averaged, quan-

tity and a prime for the fluctuating one, e.g. B = B + B′.

To determine some of the turbulent transport coefficientsin these simulations, we make use of the test-field method(Schrinner et al. 2005, 2007; Warnecke et al. 2018). This methoduses linear independent test-fields, which to do not back-react onthe flow to determine the electromotive forces of these test-fieldsusing the mean and fluctuating flow field of the simulations. Thisallows to obtain the all components of the turbulent transport ten-sors. In this work, we will only use the φφ component of the αtensor.

Some of the results, we will present in physical units byusing a normalization based on the solar rotation rate Ω⊙ =2.7 × 10−6 s−1, the solar radius R = 7 × 108 m, the density atthe bottom of the convection zone ρ(0.7R) = 200 kg/m3, andµ0 = 4π · 10−7 H m−1. Furthermore, the rotation of the simula-tions is given in terms of solar rotation rate with Ω ≡ Ω0/Ω⊙.However, the rotational influence on the convection is much bet-ter described by the use of the Coriolis number Co.

3. Results

For all the simulations we keep all input parameter constant, ex-cept that we increase the rotation rate for 0.5 to 15 solar rota-tion rate, corresponding to Co = 0.7 to 40.3. Only for the runwith the highest rotation rate (M15), we lower the diffusivities(ν, η, χSGS

m ) to keep the Reynolds and Péclet numbers on a simi-lar level, however the Prandtl numbers are kept fixed. We namethe runs with ’M’ because of their magnetic nature followed bytheir solar rotation rate. Run M5 has been discussed as Run I inWarnecke et al. (2014), as Run A1 in Warnecke et al. (2016), asRun D3 in Käpylä et al. (2017), in Warnecke et al. (2018) andas Run GW in Viviani et al. (2018). Run M3 have been analyzed

as Run B1 in Warnecke et al. (2016). Runs M10 and M15 aresimilar to Runs IW and JW of Viviani et al. (2018). As calculatedin Warnecke et al. (2018), the Rayleigh number for Run M5 isaround 100 times the critical value. We expect that this factorincreases for lower rotation and decrease for higher rotation.

In this work, we will not discuss all properties of the rota-tional influence of the hydrodynamical dynamics, i.e. the angu-lar momentum evolution, we will instead focus on the discussionand analysis of the dynamo cycles and their possible origin.

Before we do this in detail, let us look at the differen-tial rotation generated in these simulation by the interplayof rotation and turbulent convection. In Fig. 1, we showthe time averaged differential rotation Ω = Ω0 + u/r sin θfor all runs. In agreement with earlier findings (Gastine et al.2014; Käpylä et al. 2014; Fan & Fang 2014; Karak et al. 2015;Viviani et al. 2018), for slow rotation of Co = 0.7–2.9 theequator is rotating slower than the poles, so-called anti-solardifferential rotation and for rapid rotation the Co = 3.7–40the poles are rotating slower than the equator, so-called solar-like differential rotation. The overall relative latitudinal andradial shear is the strongest for the slowest rotation and de-creases in the solar-like differential regime for higher rotationrate. This agrees quantitatively with what is found in observa-tion (Reinhold et al. 2013; Lehtinen et al. 2016), with mean-fieldmodels (Kitchatinov & Rüdiger 1999) and previous global simu-lations (e.g. Käpylä et al. 2011; Gastine et al. 2014; Käpylä et al.2014; Viviani et al. 2018). The difference between the northernand southern hemisphere, for example in Runs M0.5 and M2,is caused by a hemispheric dynamo producing stronger mag-netic field in one hemisphere than in the other, see Section 3.1.An interesting feature is the occurrence of a region of a mini-mum of Ω at mid-latitude in the solar-like differential rotationregime. This region corresponds to area of large negative shearand has been shown to be responsible for the equatorward mi-grating dynamo wave in Run M5 (Warnecke et al. 2014, 2016;Käpylä et al. 2017; Warnecke et al. 2018). This region seems tobecome less pronounced for more rapidly rotating simulations.

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Fig. 1. Normalized differential rotation Ω/Ω0 with Ω = Ω0 + u/r sin θfor all runs. Ω has been calculated as a time average over the saturatedstate.

3.1. Magnetic cycles

All simulations discussed here show large-scale dynamo action.Lowering the rotation rate below Ω = 0.5, produces only aweak large-scale magnetic field (Ω = 0.4) or no dynamo action(Ω ≤ 0.3) for the same parameters otherwise. A small-scale dy-namo is not present in any of these simulations (Warnecke et al.2018). In Fig. 2, we show the near-surface mean azimuthal mag-

netic field Bφ as function of time and latitude, so-called butterflydiagram for all runs. For the slow rotating Run M0.5, we find thedynamo produces a large-scale magnetic field most pronouncedin one hemisphere and with no polarity reversals, only the am-plitude shows weak cyclic variations. For Runs M1 to M2.5 themagnetic field is of chaotic nature with polarity reversal, whichseems not to be cyclic. This is similar, what have been found be-fore for slowly rotating convective dynamos (Fan & Fang 2014;Karak et al. 2015; Hotta et al. 2016; Käpylä et al. 2017). Inter-estingly, Run M1 shows some indication of an oscillating mag-netic field with anti-solar differential rotation, however, from thecurrent running time, we cannot draw any certain conclusions.Indication of cyclic solution in the anti-solar regime have beenalso found by Karak et al. (2015), but only recently Viviani et al.(2018) have obtained clear cyclic solutions with many polar-ity cycles. Run M3 shows indication of a cyclic magnetic field,most pronounced at the poles. The magnetic cycle is even morepronounced in the middle of convection zone, as shown inWarnecke et al. (2016). The Runs M4 to M15 show a clear cyclicmagnetic field with regular polarity reversals. The cycle lengthfor these runs seems to be very similar. For Runs M4 to M10the magnetic field shows a clear equatorward propagation sim-ilar what it observed for the solar activity belt. Furthermore, ashorter, much weaker poleward migrating cycle is present in ad-dition to the equatorward migrating mode. It seems to becomestronger for increasing rotation. This weak short cycle has beenassociated with a local α2 dynamo mode in addition to the αΩdynamo causing the equatorward migration (Käpylä et al. 2016;Warnecke et al. 2018). For Run M15 the field shows indicationof both equatorward and poleward migration.

To quantify the cycle period of the magnetic field of all runs,we calculate the power spectrum of the magnetic field and usethe strongest peak as the cycle frequency. In the following wedistinguish between magnetic cycle and activity cycle. The mag-netic cycle includes a full polarity reversal, corresponding to the22 yrs on the Sun, the activity cycle uses the maximum and min-ima of the magnetic energy, so 11 yrs for the Sun. We use twoways to calculate our cycle period, for the first we determinethe magnetic cycle and take the half and for the second we deter-mine the activity cycle, both results are shown in Table 1. For themagnetic cycle we take the radial and azimuthal mean magneticfield component and calculating the power spectrum for each lat-itude and at three radii (r = 0.98, 0.85, 72). Then we average thespectra over latitude to obtain six spectra. Now, we determinethe frequency of largest speak of each spectrum with a corre-sponding error. The error is estimated by taking the full-width-half-maximum unless its narrower than the local grid spacing inthe frequency space, in which case we take local grid spacingas an error estimate. From the six frequencies, we calculate theweighted average of the corresponding periods. We take the halfof the averaged magnetic cycle period and show it as Pcycl withits error EPcycl

in Table 1.

For the activity cycle, we use the root-mean-squared value

of the magnetic field Brms =

√

Br2 + Bθ2 + Bφ2 near the surface

and as a function of radius. As above we calculate the spectrum

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Warnecke: Dynamo cycles in global convection simulation

Fig. 2. Mean azimuthal magnetic field Bφ as a function of time in years and latitude near the surface (r = 0.98R) for all runs. The time intervalshows the full duration of the saturated state for Runs M0.5 to M3 and an interval of 32 yrs for Runs M4 to M15 to illustrate the similarity in cyclelength. The black-white dashed horizontal line indicates the equator.

for each latitude and radius and averaged over them. The cy-cle period is determined in the same way as for magnetic cycle,without dividing by two. The results are shown as Pcycl with its

error EPcyclin Table 1. In Fig. 3, we show as an example the

power spectra of Run M7. Beside the cycle with an activity pe-riod of around 2.7 yr, we notice the weak short cycle, which isalso visible in Fig. 2.

For the Runs M4 to M15, the cycle periods can be deter-mined well with a small error in Pcycl. Furthermore, for these

Runs Pcycl agrees very well with Pcycl, even though their errors

are higher. The larger errors of Pcycl are caused by the summa-tion over phase-shifted magnetic field components and this can

result in a less pronounced peak in the spectrum of Brms. Thiscan be also seen in the fact that EPcycl

is for all runs significantlarger than EPcycl

. For the slowly rotating runs, the periods are notwell determined with significant differences with between the

two methods and also large errors. In the following, we there-fore focus on the analysis of the runs with a clear cycle period.

In Fig. 4 we show the cycle periods as a function of Coriolisnumber. We find two clearly separated group of runs: the slowlyrotating simulations with long ill-determined cycles (Runs M1 toM3) and the moderately to rapidly rotating runs with short cycles(Runs M4 to M15). In the latter group the cycle periods increaseslightly with increasing rotation. We perform a power law fit fortheses runs and obtain Pcycl ∝ Co0.25±0.04, or in terms of rotation

period Pcycl ∝ P−0.33±0.05rot . This value is in disagreement with

the observation of Noyes et al. (1984b) and Oláh et al. (2016),but agree qualitatively with scaling of advective dominated flux-transport dynamo models (e.g. Dikpati & Charbonneau 1999;Bonanno et al. 2002; Jouve et al. 2010). Strugarek et al. (2017)also found an increase of cycle period with increasing rotation,however their power law fit reveals a much steeper increase with

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Fig. 3. Magnetic power spectra for Run M7. (a) Spectra of the mean

radial magnetic field Br (solid) and the mean azimuthal magnetic field

Bφ (dashed) near the surface (r = 0.98; black), in the middle of theconvection zone (r = 0.85; red) and at the bottom of the convection

zone (r = 0.72; blue). (b) Spectra of Brms =

√

Br2 + Bθ2 + Bφ2 near the

surface (r = 0.85; red) and averaged over radius (black). All spectra areobtained for each latitude and then averaged. The peaks in a correspondto magnetic cycle periods and in b to activity cycles periods. The solidvertical lines indicate the cycle periods determine from the weightedaverage of the spectra in a (Pcycl), the dashed lines indicate the cycle

periods determine from the weighted average of the spectra in b (Pcycl).

Fig. 4. Cycle periods as a function of Coriolis number Co showing Pcycl

in black and Pcycl in orange. The green dashed line indicates a power lawfit of the Runs M4 to M15.

rotation Pcycl ∝ P−1.06rot . The cycle period calculated for Run M5

agrees with the cycle periods obtain for similar runs using theD2 phase dispersion statistics and the Ensemble Empirical ModeDecomposition (Käpylä et al. 2016, 2017).

3.2. Cause of magnetic cycles

Earlier studies of similar simulations as Run M5 show that theequatorward migrating mean magnetic field can be well ex-plained with a Parker-Yoshimura (Parker 1955; Yoshimura 1975)αΩ-dynamo wave propagating equatorward (Warnecke et al.2014, 2016, 2018). Following the calculation of Parker (1955)and Yoshimura (1975), we can compute the cycle frequency of

Fig. 5. Mean magnetic field, α effect and radial shear profiles forRuns M1, M3 and M7. We show the rms mean azimuthal magnetic field

averaged over the saturate state Brmsφ (top row), αφφ determined with the

test-field method (middle row) and the radial shear r sin θ∂Ω/∂r (bottomrow). The dashed lines indicate the region where we calculate PPY.

the dynamo wave using (see also Stix 1976)

ωPY =

∣

∣

∣

∣

∣

∣

αφφ kθ

2r cos θ

∂Ω

∂r

∣

∣

∣

∣

∣

∣

1/2

, (7)

where kθ is the latitudinal wave number. The corresponding ac-tivity cycle period is then given by

PPY =2π

2ωPY

. (8)

As pointed out by Warnecke et al. (2014), to get a meaningfulresult for the direction and therefore the period of dynamo wave,the location where measuring the shear and the αφφ is crucial.Following this work, we calculate PPY in the region where i)

Brmsφ = (〈B2

φ〉t)1/2 is large, in our case at least larger than the

half of the maximum value, ii) the radial shear ∂Ω/∂r is neg-ative and iii) αφφ is positive. The last two criteria are neededto excite an equatorward migrating dynamo wave, following theParker-Yoshimura sign rule. To make sure that these drivers arereally responsible to excite a dynamo wave at this location, theproduction of azimuthal magnetic field must be large at this lo-cation, leading to criterion i). The lower limit of half of the max-imum value is a reasonable choice; a slightly different value hasonly little effect on the cycle period determination and on its de-pendence on rotation rate. The criteria have been also success-fully used to confirm Paker-Yoshimura dynamo waves in simi-lar simulations (Warnecke et al. 2014, 2016, 2018; Käpylä et al.2016). Using these criteria, we then average over these regions.In Fig. 5, we show these regions for Run M1, M3 and M7. Tocalculate PPY, we choose

kθ =1

R(1 − 2Θ0/π),= 1.2/R, (9)

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Warnecke: Dynamo cycles in global convection simulation

Fig. 6. (a) Comparison of the cycle periods Pcycl (black) and Pcycl (or-ange) with predicted cycle periods using a Parker-Yoshimura dynamowave PPY (blue) for Runs M4 to M15. The dashed blue line indicatesa power law fit to PPY. (b) Contributions to the Parker-Yoshimura dy-namo wave containing the radial shear (black line; left y-axis) and αφφ(red; right y-axis) for Runs M4 to M15. The dashed lines indicate thecorresponding power law fits.

where the factor 1 − 2Θ0/π takes into account the absent of thepoles in our simulations. However, the actually value of kθ willonly affect the value PPY with a −1/2 dependency, but not thescaling with rotation.

In Table 1, we list all computed values for PPY in the eleventhcolumn and they agree well with the values of Pcycl and Pcycl

for the runs with well determined cycles (M4 to M15). For os-cillatory solution of planetary dynamos, Gastine et al. (2012)found also a good agreement between rotational dependency ofmeasured and dynamo wave predicted cycle length. In Fig. 6a,we show for these runs the cycle periods Pcycl and Pcycl to-gether with the predicted period PPY. A power-law fit results inPPY ∝ Co0.51±0.05 which is close to Pcycl ∝ Co0.25±0.04. Therefore,the Parker-Yoshimura dynamo wave explains well the weaklydependency of cycle frequency with rotation, we find for themoderately and rapidly rotating simulations. We now go a stepfurther and check, which mechanism of dynamo wave causesthis rotational dependency. For this we plot in Fig. 6b the ro-tational dependency of the radial shear and α effect in termsof |r cos θ|∂Ω/∂r and αφφ; as for PPY both quantities are av-eraged over the region of interest. The strength of the shearstrongly weakens for larger rotation, with an estimated scalingof Co−1.33±0.18. For Run M15 the shear in the region is just be-low zero explaining the mixture of equatorward and polewardmigration pattern shown in Fig. 2. For αφφ, we find an increase

with rotation corresponding to a scaling of Co0.70±0.25, so muchless than linear. The strong decrease in shear causes the cyclesto become larger with rotation: assuming a constant αφφ, shear

alone would leading a scaling of Pcycl ∝ Co0.67±0.09. The α-effecton the other hand lead to a decrease of cycle length with rota-tion; Pcycl ∝ Co−0.35±0.12. Because the increase of cycle lengthdue to shear is stronger than the decrease due to the α effect, theresulting cycle length shows only a weak increase with rotation.

Fig. 7. Ratio of rotation period and cycle period Prot/Pcycl over theCoriolis number Co. The black asterisks indicate Pcycl and the orange

ones Pcycl. The blue solid line shows the predicted cycle length fromEquations (7) and (8), the green dashed line the fit of Fig. 4 and the reddashed line a power law fit of Runs M4 to M15.

The surprising issue with interpretation of the magnetic fieldevolution as a Parker-Yoshimura dynamo wave is that for runsrotating slower than the M4 (Co = 6.5) it fails. Equation (7) forthese runs predict cycle periods of similar length as for the morerapidly rotating runs, but the actual magnetic field shows no clearcyclic evolution. For example, Run M3 shows similar conditionfor a dynamo wave as in Run M7; there exists a localized region,where the mean toroidal field is strong, αφφ is positive and shearnegative, see Fig. 5. In the simulations with anti-solar differentialrotation (Run M0.5 to M2), we find instead a more extendedregion of strong mean toroidal field, positive αφφ and negativeshear, however strength of the shear and the α effect should besufficient to excite an αΩ dynamo wave. One of the reasons forthe absent of an αΩ dynamo wave can be the larger turbulentmagnetic diffusion due to higher convective velocities as shownin Fig. 9. This is in agreement with previous studies of rotatingconvection in Cartesian boxes (Käpylä et al. 2009). To have areliable statement, if a αΩ dynamo is actually operating in thesesimulation, and what is the reason for not exciting dynamo wave,need to be studied in more detail using all the turbulent transportcoefficients. We postpone such a study to the future.

We can now also interpret the scaling of the shear and theα effect in terms of mean-field models (e.g. Krause & Rädler1980; Rüdiger 1989). From models of differential rotation, onetypically finds that the absolute radial and latitudinal differen-tial rotation stays nearly constant for increasing rotation (e.g.Kitchatinov & Rüdiger 1999), which disagrees with our find-ings. However, we stress here that in these models they take thelatitudinal averaged values at bottom and at surface to computethe radial differential rotation, we compute the local radial shearin the region of interest. In mean-field dynamo models αφφ is

related to the mean kinetic helicity u′∇ × u′ and therefore is lin-ear related to the Ω. Taking also the convective turnover time τc

into account leads to scaling of αφφ ∝ Co (e.g. Krause & Rädler1980). As shown in Warnecke et al. (2018), approximating the

diagonal α components with αK = −1/3τc u′∇ × u′ is not cor-rect and can lead to the overestimation of αφφ. Indeed, we findαφφ in the region of interest depends weaker on rotation as pre-

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Fig. 8. Ratio of rotation period and cycle period Prot/Pcycl over mag-netic and kinetic energy ratio. The black asterisks indicate Pcycl and the

orange ones Pcycl.The red dashed line indicates a power law fit of Run sM4 to M15.

dicted from mean-field models. The overall scaling of αφφ av-eraged of the simulations might be different, but the importvalue of αφφ determining the cycle period comes from this re-gion. Warnecke et al. (2018) took also into account the non-linear quenching of the α effect due to magnetic helicity con-servation (see Brandenburg & Subramanian 2005, for details)and use the form introduced by Pouquet et al. (1976) α =

−1/3τc u′ · ∇ × u′ + 1/3τc/ρ∇ × b′ · b′, but still could not finda agreement with the actual measured αφφ, see Fig. 1 and 2 ofWarnecke et al. (2018).

Furthermore, we plot the ratio of rotation period and cycleperiod Prot/Pcycl over Coriolis number, see Fig. 7. We find a scal-

ing of Prot/Pcycl ∝ Co−0.98±0.04 for the runs with well determinedcycle. This scaling fits well with the cycle period predicted by adynamo wave. Interestingly, Run M0.5 fit well to this relation,even though we find no polarity reversals there. In interpreta-tion of stellar observation, Prot/Pcycl is often used to determinethe quenching of the α effect. If one assumes a linear depen-dency of α and ∂Ω/∂r onΩ together with Equation (7), Prot/Pcycl

over Co give an estimate over the rotational quenching of α (e.g.Brandenburg et al. 1998). Moreover, one can go a step furtherand plot Prot/Pcycl over magnetic activity, which is related to thesurface magnetic field strengths (e.g. Schrijver et al. 1989). Incase of dynamo simulations we can instead use the ratio of mag-netic and kinetic energy, so-called dynamo efficiency, to mimicthe magnetic activity as done in Fig. 8. Then the Prot/Pcycl de-pendence on magnetic activity can be interpreted as the magneticquenching of the α effect. However, doing so stellar observa-tions indicate an increase of the α effect with magnetic activ-ity in the inactive and active branch (Brandenburg et al. 1998;Saar & Brandenburg 1999; Brandenburg et al. 2017).

In our simulations, the situation is different. As describedabove, the radial shear decreases and the α effect increases withhigher rotation rate. Therefore, we cannot use Prot/Pcycl to esti-mate the quenching of the α effect. The scaling of Prot/Pcycl ∝

Co−0.99±0.04 as shown in Fig. 7 might seem to be expected, be-cause we plot rotation rate over rotation rate, but the Corio-lis number includes also the strength of convection, in termsof τc, which is influenced by rotation as well. We do not find

Fig. 9. Turbulent (eddy) magnetic diffusivity over Coriolis number Co.The diffusivity is determined using an estimate of the turbulence in thesimulations ηt = 1/3urms/kf (black) and using the cycle periods as inEquation (12) (red) and as in Equation (13) following Roberts & Stix(1972) (blue). The green dashed line indicates a power law fit of theblack asterisk of Runs M4 to M15.

any indication of branches with positive slopes similar to theinactive or active branch as postulated by Brandenburg et al.(1998). Furthermore, also the slope is different from what isfound from the super-active branch, which has a slope of Co−0.43

(Saar & Brandenburg 1999).In Fig. 8, we plot Prot/Pcycl over ratio of magnetic and kinetic

energy, which can be interpreted as the dynamo efficiency, wefind a scaling of Prot/Pcycl ∝ (Emag/Ekin)−1.29±0.05. The scalingseems to agree qualitatively with what is found in Viviani et al.(2018). Also, here, we do not find any indication of a posi-tive slope and therefore a similarity to the activity branches. Itseems clear the runs with well determined cycles cannot be in-terpreted in terms of activity branches with positive slopes. How-ever, our scaling agrees qualitatively with the suggested transi-tional branch by Distefano et al. (2017).

Another way to analyze the scaling of the cycle period withrotation is via the turbulent eddy magnetic diffusivity. In the sat-urated state the dynamo drivers has to balance to contribution ofthe magnetic diffusion. In a αΩ dynamo wave, the balance reads

ωPY − k2ηt = 0, (10)

where k is a wavenumber and ωPY is given by Equation (7).Therefore, we can use this equation to calculate ηt based onthe cycle frequency and compare with estimated values us-ing the turbulent flow of the simulations. Using the first-oder-smoothing-approximation (FOSA, see e.g. Krause & Rädler1980) and isotropic and homogeneous turbulence, the turbulenteddy diffusivity can be estimated as

ηt =1

3

urms

kf

. (11)

If we assume k = kθ in Equation (10), we can calculate ηt basedon the cycle period

ηt =2π

2 Pcycl k2θ

. (12)

Another way is to use the radius R and the depth of the con-vection zone 0.3R of the star to relate ηt with the cycle period(Roberts & Stix 1972).

ηt =0.3R2

2 Pcycl

. (13)

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We show the values for both expressions for the runs with welldetermine cycles (Runs M4 to M15) together with values ofEquation (11) as a function of Coriolis number in Fig. 9. ηt ofEquation (13) fits remarkable well with Equation (11). Even thescaling of ηt = Co−0.27±0.02 determined from Equation (11) isthe same as expected from the cycle periods ηt ∝ 1/Pcycl ∝

Co−0.25±0.04, see Fig. 4. This good agreement is indeed interest-ing and not fully expected, because the estimation of ηt in Equa-tion (11) is based on strong assumptions, which are most likelynot fulfilled in these simulations. The values obtained throughEquation (12) have obviously the same scaling as Equation (13),however the values are around a factor of 10 higher. This is be-cause of the different values of scales/wave numbers included inthis calculation. If we use kθ instead of kf in Equation (11), thecurves would lie closer together. Therefore, the scaling of cycleperiod with rotation rate of Pcycl ∝ Co0.25±0.04 can also be wellexplained with the rotational quenching of the turbulent (eddy)magnetic diffusivity.

3.3. Comparison with observational and other numericalstudies

There is only limited amount of numerical studies of cyclesof solar and stellar dynamos. In the following, we compareour results with the recent work of Strugarek et al. (2017) andViviani et al. (2018). The study by Strugarek et al. (2017) in-clude seven models, all showing cyclic dynamo solutions. Eventhough the authors interpret their models in the vicinity of theSun, their Rossby numbers indicate a rapid rotational regime. Ifwe convert their numbers to our definition of Coriolis numbers orinverse Rossby number as defined in Brandenburg et al. (1998),respectively, we obtain values of Co = 18 to 62. These highnumbers are due to their low convective velocities compared toother models (Käpylä et al. 2017). For comparison the estimatedCoriolis number of the Sun is Co = 6.2 (e.g. Brandenburg et al.2017). In Fig. 10, we plot the models of Strugarek et al. (2017)together with our models. There seems to be no overlap be-tween their and our simulations. However, also their simula-tions show a decrease of Prot/Pcycl with rotation following aneven steeper slope. The similar slope might be caused by a sim-ilar dynamo mechanism and the small difference and the off-set might be because of different system parameters, as Prandtlnumbers and/or Reynolds numbers. As shown in Käpylä et al.(2017), the Reynolds numbers in typical simulations with theEULAC code can be lower compared to models of other, sim-ilar codes. This might also explain the dominantly axisymmet-ric large-scale magnetic field solution in Strugarek et al. (2017).The simulations of Viviani et al. (2018) show clearly that if thereis a transition from axisymmetric to non-axisymmetric mag-netic solution at Co ≥ 3. However, if the resolution, thereforethe Reynolds and Rayleigh numbers, are not high enough, thenon-axisymmetric magnetic solution can be not obtained. Thisis most important for large rotation rates, where the convectionis rotationally quenched. Furthermore, Strugarek et al. (2017)claim that their dynamo cycle is caused by a non-linear feed-back of the torsional oscillation on the magnetic field. Such aninterpretation is not very likely to be correct, because to have acyclic torsional oscillation one needs a cyclic dynamo in the firstplace. In the light of the results of this work we are inclined tothink that also their cyclic magnetic field is caused by a Parker-Yoshimura dynamo wave. Indeed, their differential rotation pro-files show localized regions of strong negative shear, where alsothe mean magnetic field propagates equatorward.

The study of Viviani et al. (2018) probe a large range ofrotation rate in particular in the rapid rotational regime. Theirsimulations using a similar setup than the one in this work,but for most of their simulations they use a full 2π extend inthe azimuthal direction (2π runs) and obtain non-axisymmetriclarge-scale field solutions for moderately to rapidly rotatingruns. In Fig. 10, we also over-plot their simulations. Their π/2wedge runs agrees well with our runs and our obtained scal-ing of Co−0.98±0.04. This also means that their estimates of cy-cle periods based on the temporal variation in the large-scalemagnetic energy seems to describe the magnetic cycle well.Their 2π runs with rapid rotation (Co ≥ 3) show clearly adifferent scaling, similar to Co−0.43 of the super-active branch(Saar & Brandenburg 1999; Viviani et al. 2018). This mightmean that the non-axisymmetric magnetic field solutions havea different scaling than the pure axisymmetric ones. The argu-ment of different scaling is also supported from the fact that theslowly rotating axisymmetric simulations of Viviani et al. (2018)can be well described by the cycle period scaling estimated inthis work, see Fig. 10. However, surprising is the fact that fornearly the same rotation rate the 2π runs show much shorter andmuch clearer cycles than their corresponding π/2 wedge simula-tions.

At the end, we compare our results with observational ob-tained stellar cycles. One sample comes from Lehtinen et al.(2016), where the authors use photometry to measure cyclic vari-ation is solar-like stars. From this sample, we only plot the cy-cles which are identified as better than “poor” and plot themin Fig. 10. For rapid rotating stars, their cycles fall surprisingwell on our scaling relation even though their magnetic fieldis non-axisymmetric. We note here that Lehtinen et al. (2016)did not do any direct measurement of the magnetic field, butinferred the degree of non-axisymmetry from the spot distribu-tions. For slowly rotation, the stars seems to fall on a parallelline with a similar scaling. Lehtinen et al. (2016) found that thecrossover from the transitional branch to the super-active branchhappens at around Co = 26.3 and chromospheric activity valueof log R′

HK= −4.4. We further include the sub-sample of stars

from the Mount Wilson sample analyzed by Brandenburg et al.(2017). Our scaling relation falls through these stars; however,a lot of stars are not captured by our scaling. As shown byBrandenburg et al. (2017), the stars around Co = 10 form theinactive branch with a positive slope and the stars with higherrotation form the active branch also with a positive slope. Theactive branch is not as confined as the inactive one. In recentworks the existence of these branches have been questioned(Reinhold et al. 2017; Distefano et al. 2017). Two new stud-ies of the Mount Wilson sample data (Boro Saikia et al. 2018;Olspert et al. 2017) only find an indication for an inactive branchand otherwise a distribution similar to our scaling showing indi-cation of an transitional branch. There, the authors re-analysesthe full Mount Wilson sample, without relying of the cycle deter-mination of Baliunas et al. (1995). The problem with determin-ing cycles from chromospheric and photospheric activity timeseries is due to the method to for the period search. For star in theinactive branch the cycles are clean and can be well determined,but more active stars with higher rotation rates exhibit a complexbehavior with multiple cycles (e.g. Oláh et al. 2016). There, thecalculated cycle period can differ depending with method is used(e.g. Olspert et al. 2017). We also note that one to one compar-ison with observation is not always possible. Even though us-ing the Coriolis number is more meaningful than then the ro-tational period, because it indicates if the star or simulation isin the slow or rapidly rotating regime, the convective turnover

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Fig. 10. Ratio of rotation period and cycle period Prot/Pcycl over the Coriolis number Co. The black asterisks indicate Pcycl. The dashed red

line indicates the fit Co−0.98 of Fig. 7 and the dashed black line the transition from anti-solar to solar like differential rotation. We include thesimulations of Viviani et al. (2018) (blue squares, with crosses for wedge runs) and Strugarek et al. (2017) (green diamond), the observationalstudies of Lehtinen et al. (2016) (purple crosses) and of Brandenburg et al. (2017) (light grey triangles for K dwarfs, dark grey circles for F, Gdwarfs, including the Sun: yellow cross).

time is usually ill determined for observed stars. Furthermore,as we know the turnover time and therefore the correspondingCoriolis number in the Sun changes several orders of magnitudefrom the solar surface to the bottom of convection zone (e.g. Stix2002), it is difficult even for the Sun to estimate a single numberas a meaningful Coriolis number. Therefore, Brandenburg et al.(2017) and Olspert et al. (2017) use the chromospheric activ-ity instead of the Coriolis number for their analysis. Using this,Olspert et al. (2017) finds that the cycle periods of this work fitvery well with their observed stellar cycles, see their Fig. 6.

Interestingly the cycle data taking from Brandenburg et al.(2017) indicate that the Sun lies close to our scaling relation, ac-tually close to Run M4. If we go a step further and assume theSun’s 22 yrs magnetic cycle is caused by a Parker-Yoshimuradynamo wave with a cycle-rotation scaling similar to our simu-lations, we can estimate the corresponding Coriolis number andthen with the definition of Equation (5) we also calculate thecorresponding value of urms. We calculate a Coriolis number ofCo = 8.5 corresponding to urms = 21.5 m/s. This velocity wouldbe located at around 160 Mm depth (r = 0.78 R) according tothe mixing length model of Spruit (1974) or r = 0.72 R for themodel of Stix (2002). Therefore, this kind of dynamo wave can-not be driven by the near-surface shear layer, it must instead bedriven by a positive radial shear and an inversion of sign of αφφin the deeper part of the convection zone to get an equatorwardmigrating magnetic field (Duarte et al. 2016).

4. Conclusions

We use 3D MHD global dynamo simulations to investigate therotational dependency of magnetic activity cycles. For moder-ately and rapidly rotating runs (Co ≥ 6.5), we find well-definedcycles in range between 2 to 4 yrs. For slowly rotating runs, wefind irregular cycles with mostly longer periods. There the cycleperiods can be only ill-determined. Using the φφ component ofthe α tensor measured with the test-field method, and the radialshear we find a good agreement of the cycle period predicted by aParker-Yoshimura dynamo wave for moderately to rapid rotatingruns. There we find that the cycle period only weakly dependson rotation (Pcycl ∝ Co0.25±0.04 and Pcycl ∝ P−0.33±0.05

rot ). Also thisscaling is well reproduced by a Parker-Yoshimura dynamo wave.αφφ increases only weakly with rotation (αφφ ∝ Co0.70±0.25) andthe strength of negative radial shear decreases larger than lin-ear with rotation ∂Ω/∂r ∝ Co−1.33±0.18. This is not in agreementwith mean-field theory, where α depends linear on the rotation(e.g. Krause & Rädler 1980). Also from models of differentialrotation ones finds the radial shear has not a strong dependencyon rotation (Kitchatinov & Rüdiger 1999). However, these mod-els usually look at the global quantities and we determine ourscaling from localized region responsible for driving the dynamowave.

Looking at the ratio of rotation and cycle period over Cori-olis number, we do not find any indication of activity brancheswith positive slopes as found from observation of stellar cycles

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Warnecke: Dynamo cycles in global convection simulation

by Brandenburg et al. (1998) and Saar & Brandenburg (1999).The negative slope of our simulations with well determinedcycles seems to be more in agreement with the transitionalbranch postulated by Distefano et al. (2017) and confirmed byBoro Saikia et al. (2018) and Olspert et al. (2017). Furthermore,our results suggest that the cyclic magnetic fields found in thework by Strugarek et al. (2017) are also caused by a Parker-Yoshimura dynamo wave, because indeed their simulations pro-duce strong negative shear in the location, where the magneticfield is oscillating. By assuming the solar magnetic cycle iscaused by a Parker-Yoshimura dynamo wave following a similarscaling as in our simulations, we can conclude that the dynamoin the Sun operates near the bottom of convection zone, whereturbulent velocities are around 20 m/s.

The cycle period dependence on rotation of our simulationscan be also well explained via the rotational quenching of theturbulent diffusivity, which contribution has to balance with thedynamo driver in the saturated stage. We find the nearly the samescaling for the turbulent eddy diffusivity with Coriolis number asexpected from a direct inverse proportionality of the diffusivitywith cycle period (e.g. Roberts & Stix 1972).

For the slowly rotating runs a Parker-Yoshimura dynamowave seems to be not excited, as the predicted periods do not fitwith the measured ones. This might be due to a higher turbulentdiffusion in this rotation regime as found in Käpylä et al. (2009).In particular, these simulations need to be further investigated us-ing the full set of turbulent transport coefficients. Moreover, wenote here that in the rapidly rotating regime the magnetic fieldcan become highly non-axisymmetric as found recently in obser-vations (e.g. Lehtinen et al. 2016) and simulations (Viviani et al.2018). The work of Viviani et al. (2018) indicate a weaker scal-ing of Prot/Pcycl with Coriolis number than what we find in ourwork and this might be due to the non-axisymmetric magneticfield solution, which are suppressed in our work because of thewedge assumption.

We stress here that the cycles determined from our simu-lation are directly linked to the magnetic field. Observationalstellar cycles are mostly measured from chromospheric activ-ity variations (Ca II H&K) or photometry. These are only prox-ies of the magnetic field strength and might not capture all thefeatures of cyclic variations. Therefore, it would be useful to de-termine also the integrated variability caused by the simulatedmagnetic cycles. Furthermore, we will in future also investigatehow coronal heating and therefore the X-ray luminosity will de-pend on the cyclic magnetic field. For this it is crucial to com-bine convective dynamo models with a coronal envelope as donein Warnecke et al. (2011, 2012, 2013, 2016). This is in particularimportant to study the role of helicity connecting the dynamo ac-tive stellar convection zones with stellar coronae, as the magnetichelicity might play an important role in the heating of coronae(Warnecke et al. 2017).

Acknowledgements. We thank the referee Günther Rüdiger and our colleaguesMaarit J. Käpylä, Mariangela Viviani and Jyri J. Lehtinen for comments on themanuscript and discussion leading to this work. The simulations have been car-ried out on supercomputers at GWDG, on the Max Planck supercomputer atRZG in Garching, in the facilities hosted by the CSC—IT Center for Sciencein Espoo, Finland, which are financed by the Finnish ministry of education. .W.acknowledges funding by the Max-Planck/Princeton Center for Plasma Physicsand from the People Programme (Marie Curie Actions) of the European Union’sSeventh Framework Programme (FP7/2007-2013) under REA grant agreementNo. 623609.

References

Augustson, K., Brun, A. S., Miesch, M., & Toomre, J. 2015, ApJ, 809, 149Baliunas, S. L., Donahue, R. A., Soon, W. H., et al. 1995, ApJ, 438, 269Barekat, A., Schou, J., & Gizon, L. 2014, A&A, 570, L12Basu, S. 2016, Living Reviews in Solar Physics, 13, 2Beaudoin, P., Simard, C., Cossette, J.-F., & Charbonneau, P. 2016, ApJ, 826, 138Bonanno, A., Elstner, D., Rüdiger, G., & Belvedere, G. 2002, A&A, 390, 673Boro Saikia, S., Marvin, C. J., Jeffers, S. V., et al. 2018, A&A, accepted

[arXiv:1803.11123]Brandenburg, A. 2005, ApJ, 625, 539Brandenburg, A., Mathur, S., & Metcalfe, T. S. 2017, ApJ, 845, 79Brandenburg, A., Saar, S. H., & Turpin, C. R. 1998, ApJ, 498, L51Brandenburg, A. & Subramanian, K. 2005, Phys. Rep., 417, 1Brown, B. P., Browning, M. K., Brun, A. S., Miesch, M. S., & Toomre, J. 2008,

ApJ, 689, 1354Charbonneau, P. 2014, ARA&A, 52, 251Dikpati, M. & Charbonneau, P. 1999, ApJ, 518, 508Distefano, E., Lanzafame, A. C., Lanza, A. F., Messina, S., & Spada, F. 2017,

A&A, 606, A58Duarte, L. D. V., Wicht, J., Browning, M. K., & Gastine, T. 2016, MNRAS, 456,

1708Fan, Y. & Fang, F. 2014, ApJ, 789, 35Gastine, T., Duarte, L., & Wicht, J. 2012, A&A, 546, A19Gastine, T., Yadav, R. K., Morin, J., Reiners, A., & Wicht, J. 2014, MNRAS,

438, L76Gent, F. A., Käpylä, M. J., & Warnecke, J. 2017, Astronomische Nachrichten,

338, 885Ghizaru, M., Charbonneau, P., & Smolarkiewicz, P. K. 2010, ApJ, 715, L133Gilman, P. A. 1983, ApJS, 53, 243Hanasoge, S., Gizon, L., & Sreenivasan, K. R. 2016, Annual Review of Fluid

Mechanics, 48, 191Hotta, H., Rempel, M., & Yokoyama, T. 2016, Science, 351, 1427Jouve, L., Brown, B. P., & Brun, A. S. 2010, A&A, 509, A32Käpylä, M. J., Käpylä, P. J., Olspert, N., et al. 2016, A&A, 589, A56Käpylä, P. J., Käpylä, M. J., & Brandenburg, A. 2014, A&A, 570, A43Käpylä, P. J., Käpylä, M. J., Olspert, N., Warnecke, J., & Brandenburg, A. 2017,

A&A, 599, A4Käpylä, P. J., Korpi, M. J., & Brandenburg, A. 2009, A&A, 500, 633Käpylä, P. J., Mantere, M. J., & Brandenburg, A. 2011, Astron. Nachr., 332, 883Käpylä, P. J., Mantere, M. J., & Brandenburg, A. 2012, ApJ, 755, L22Käpylä, P. J., Mantere, M. J., Cole, E., Warnecke, J., & Brandenburg, A. 2013,

ApJ, 778, 41Karak, B. B., Käpylä, P. J., Käpylä, M. J., et al. 2015, A&A, 576, A26Kitchatinov, L. L. & Rüdiger, G. 1999, A&A, 344, 911Kleeorin, N. I., Ruzmaikin, A. A., & Sokolov, D. D. 1983, Ap&SS, 95, 131Köhler, H. 1970, Sol. Phys., 13, 3Krause, F. & Rädler, K.-H. 1980, Mean-field Magnetohydrodynamics and Dy-

namo Theory (Oxford: Pergamon Press)Küker, M., Rüdiger, G., & Schultz, M. 2001, A&A, 374, 301Lehtinen, J., Jetsu, L., Hackman, T., Kajatkari, P., & Henry, G. W. 2016, A&A,

588, A38Noyes, R. W., Hartmann, L. W., Baliunas, S. L., Duncan, D. K., & Vaughan,

A. H. 1984a, ApJ, 279, 763Noyes, R. W., Weiss, N. O., & Vaughan, A. H. 1984b, ApJ, 287, 769Oláh, K., Kovári, Z., Petrovay, K., et al. 2016, A&A, 590, A133Olspert, N., Lehtinen, J. J., Käpylä, M. J., Pelt, J., & Grigorievskiy, A. 2017,

A&A, submitted [arXiv:1712.08240]Ossendrijver, M. 2003, A&A Rev., 11, 287Parker, E. N. 1955, ApJ, 122, 293Pouquet, A., Frisch, U., & Léorat, J. 1976, J. Fluid Mech., 77, 321Reinhold, T., Cameron, R. H., & Gizon, L. 2017, A&A, 603, A52Reinhold, T., Reiners, A., & Basri, G. 2013, A&A, 560, A4Roberts, P. H. & Stix, M. 1972, A&A, 18, 453Rüdiger, G. 1989, Differential Rotation and Stellar Convection. Sun and Solar-

type Stars (Berlin: Akademie Verlag)Rüdiger, G., Kitchatinov, L. L., Küker, M., & Schultz, M. 1994, Geophysical

and Astrophysical Fluid Dynamics, 78, 247Saar, S. H. & Brandenburg, A. 1999, ApJ, 524, 295Schou, J., Antia, H. M., Basu, S., et al. 1998, ApJ, 505, 390Schrijver, C. J., Cote, J., Zwaan, C., & Saar, S. H. 1989, ApJ, 337, 964Schrinner, M., Rädler, K.-H., Schmitt, D., Rheinhardt, M., & Christensen, U.

2005, Astron. Nachr., 326, 245Schrinner, M., Rädler, K.-H., Schmitt, D., Rheinhardt, M., & Christensen, U. R.

2007, Geophys. Astrophys. Fluid Dyn., 101, 81Spruit, H. C. 1974, Sol. Phys., 34, 277Steenbeck, M., Krause, F., & Rädler, K.-H. 1966, Zeitschrift Naturforschung Teil

A, 21, 369Stix, M. 1976, in IAU Symposium, Vol. 71, Basic Mechanisms of Solar Activity,

ed. V. Bumba & J. Kleczek, 367

Article number, page 11 of 12

A&A proofs: manuscript no. paper

Stix, M. 2002, The sun: an introduction (Springer, Berlin)Strugarek, A., Beaudoin, P., Charbonneau, P., Brun, A. S., & do Nascimento,

J.-D. 2017, Science, 357, 185Vaughan, A. H. & Preston, G. W. 1980, PASP, 92, 385Viviani, M., Warnecke, J., Käpylä, M. J., et al. 2018, A&A, accepted

[arXiv:1710.10222]Warnecke, J., Brandenburg, A., & Mitra, D. 2011, A&A, 534, A11Warnecke, J., Chen, F., Bingert, S., & Peter, H. 2017, A&A, 607, A53Warnecke, J., Käpylä, P. J., Käpylä, M. J., & Brandenburg, A. 2014, ApJ, 796,

L12Warnecke, J., Käpylä, P. J., Käpylä, M. J., & Brandenburg, A. 2016, A&A, 596,

A115Warnecke, J., Käpylä, P. J., Mantere, M. J., & Brandenburg, A. 2012, Sol. Phys.,

280, 299Warnecke, J., Käpylä, P. J., Mantere, M. J., & Brandenburg, A. 2013, ApJ, 778,

141Warnecke, J., Rheinhardt, M., Tuomisto, S., et al. 2018, A&A, 609, A51Yoshimura, H. 1975, ApJ, 201, 740

Article number, page 12 of 12